Non-perturbative selection rules in F-theory

TL;DR

This work develops a framework to distinguish perturbative and non-perturbative selection rules for charged-matter couplings in 4D F-theory by analyzing fibral curves and their homology in the Calabi–Yau fourfold. It shows that perturbative couplings arise when fibral curves sum to zero in fibre homology, while non-perturbative couplings are mediated by M2-instantons on 3-chains with finite volume, corresponding to massive $U(1)$s or discrete symmetries; the weak-coupling Type IIB picture is captured by D1-F1 instantons, with D1 contributions controlled by axionic gauging. The paper develops the resolution-homology and Mayer-Vietoris tools to connect local fibre structures to global base geometry, illustrating the ideas through simple local models and a global $SU(5)$ example, and discusses how torsion and discrete symmetries arise via the universal coefficient theorem. It also provides a concrete global-realization of M2/D1 instantons in a split $I_1$ model, highlighting how the 3-chain geometry and brane intersections govern which operators are perturbatively allowed or exponentially suppressed, with implications for string-model building and weak-c coupling limits.

Abstract

We discuss the structure of charged matter couplings in 4-dimensional F-theory compactifications. Charged matter is known to arise from M2-branes wrapping fibral curves on an elliptic or genus-one fibration Y. If a set of fibral curves satisfies a homological relation in the fibre homology, a coupling involving the states can arise without exponential volume suppression due to a splitting and joining of the M2-branes. If the fibral curves only sum to zero in the integral homology of the full fibration, no such coupling is possible. In this case an M2-instanton wrapping a 3-chain bounded by the fibral matter curves can induce a D-term which is volume suppressed. We elucidate the consequences of this pattern for the appearance of massive U(1) symmetries in F-theory and analyse the structure of discrete selection rules in the coupling sector. The weakly coupled analogue of said M2-instantons is worked out to be given by D1-F1 instantons. The generation of an exponentially suppressed F-term requires the formation of half-BPS bound states of M2 and M5-instantons. This effect and its description in terms of fluxed M5-instantons is discussed in a companion paper.

Figures (9)

• Figure 1: 3-chain $\Gamma$ connecting $\mathbb P^1_A$ and $\mathbb P^1_B$.
• Figure 2: Fibre topology at the Yukawa point $p_2 = \{w=0\} \cap \{a_1 =0 \} \cap \{a_{32}=0\}$ in the $SU(5)$ Tate model.
• Figure 3: Fibre enhancement over the Yukawa points in the $U(1)$ model introduced in Morrison:2012ei. The red and blue crosses denote intersections with the sections ${ U}$ and, respectively, $S$, which also wraps $F_c$ and $A_I$ as indicated. The self-intersection of ${\cal C}_{II}$ at $p$ is responsible for the existence of the two splitting paths (\ref{['pertCIIa']}) and (\ref{['pertCIIb']}) for the fibral curves.
• Figure 4: 3-chain $\Gamma_{\rm p}$ connecting $\tilde{A}_{II}$ and $\tilde{B}_{II}$.
• Figure 5: $T$-monodromies in the fibre obstruct the uplift of a Euclidean D1-brane intersecting a D7-brane.